Finite Group Databasebeta

← C83×#61404 #35289228

Subgroups — C83×#61404 #35289228

Order 21248.

1900 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3480C211yesmaximal, characteristic
2C2×#557×C83C211yesmaximal, characteristic, complemented
3C83×#3400C216yesmaximal, complemented
4C83×#3398C216yesmaximal, complemented
5C83×#3395C212yesmaximal, complemented
6C83×#3392C212yesmaximal, complemented
7C83×#3152C213yesmaximal, complemented
8C83×#3039C213yesmaximal, complemented
9C83×#3037C213yesmaximal, complemented
10Q8×M16×C83C211yesmaximal, characteristic, complemented
11C4×#137×C83C213yesmaximal, complemented
12C2×#137×C83C2213yes—
13C2×#137×C83C2216yes—
14#559×C83C2213yes—
15#557×C83—21no—
16#557×C83—23no—
17#557×C83C2212yes—
18#557×C83C2211yescharacteristic
19#557×C83C2216yes—
20#556×C83C2213yes—
21#553×C83—23no—
22#550×C83C2211yescharacteristic
23#548×C83—21no—
24#532×C83C2213yes—
25C2×C4⋊Q8×C83C2213yes—
26#529×C83—23no—
27C2×#125×C83—23no—
28C4×#41×C83—21no—
29C4×#41×C83C2213yes—
30C2×C4×Q8×C83—21no—
31C2×C4×Q8×C83C2213yes—
32C2×C4×Q8×C83C2211yescharacteristic
33#497×C83C22112yes—
34#494×C83C2216yes—
35#492×C83C2216yes—
36#464×C83C22112yes—
37#462×C83C2216yes—
38#462×C83C2216yes—
39#462×C83C2216yes—
40#461×C83C2216yes—
41#461×C83C2216yes—
42#460×C83C2216yes—
43#446×C83C2216yes—
44C8×Q8×C83C2212yes—
45#441×C83C2216yes—
46#440×C83C2216yes—
47C4×Q16×C83C2216yes—
48C4×SD16×C83C2216yes—
49#428×C83C2211yescharacteristic
50#423×C83C2213yes—
51#419×C83C2213yes—
52#417×C83C2213yes—
53C4×M16×C83C2213yes—
54C2×#41×C83—21no—
55C2×#41×C83C2313yes—
56C22×Q8×C83D411yescharacteristic
57C22×Q8×C83C2313yes—
58#137×C83—212no—
59#137×C83—412no—
60C2×Q16×C83C2316yes—
61C2×Q16×C83C23112yes—
62C2×SD16×C83C2316yes—
63C2×SD16×C83C23112yes—
64C2×M16×C83C2313yes—
65C2×M16×C83C2311yescharacteristic
66C4⋊Q8×C83—23no—
67C4⋊Q8×C83—23no—
68C4⋊Q8×C83C2316yes—
69C4⋊Q8×C83C2313yes—
70C4⋊Q8×C83C2316yes—
71#127×C83C2313yes—
72#126×C83—26no—
73#125×C83—26no—
74#125×C83—26no—
75#125×C83—212no—
76#125×C83—23no—
77#124×C83—23no—
78#124×C83C2316yes—
79#123×C83—26no—
80C22⋊Q8×C83—23no—
81#121×C83—23no—
82#121×C83—23no—
83#121×C83—23no—
84C4×Q8×C83—26no—
85C4×Q8×C83—22no—
86C4×Q8×C83—21no—
87C4×Q8×C83—23no—
88C4×Q8×C83—23no—
89C4×Q8×C83D411yescharacteristic
90C4×Q8×C83D413yes—
91C4×Q8×C83C2316yes—
92C4×Q8×C83C2316yes—
93C4×Q8×C83C2313yes—
94C4×Q8×C83C2313yes—
95C4×Q8×C83C2311yescharacteristic
96C4×Q8×C83C2313yes—
97C4×D4×C83—23no—
98C4×D4×C83—23no—
99C4×D4×C83—23no—
100C4×D4×C83—23no—
101C4×D4×C83—23no—
102C4×D4×C83C2313yes—
103C4×D4×C83C2313yes—
104#117×C83—23no—
105#117×C83—23no—
106#117×C83C2311yescharacteristic
107C2×C4⋊C4×C83—23no—
108C2×C4⋊C4×C83—23no—
109C2×C4⋊C4×C83—23no—
110C2×C4⋊C4×C83D413yes—
111C2×C4⋊C4×C83C2311yescharacteristic
112C2×C42×C83—23no—
113C2×C42×C83—23no—
114C2×C42×C83C2313yes—
115#107×C83C2312yes—
116#106×C83C2312yes—
117C4⋊C8×C83C2316yes—
118C4⋊C8×C83C2316yes—
119#103×C83C2316yes—
120#103×C83C2316yes—
121#103×C83C2316yes—
122#102×C83C2316yes—
123#97×C83C2316yes—
124C4×C8×C83C2316yes—
125#41×C83—26no—
126#41×C83—26no—
127#41×C83—44no—
128C2×Q8×C83—24no—
129C2×Q8×C83—23no—
130C2×Q8×C83—26no—
131C2×Q8×C83—26no—
132C2×Q8×C83C2×D413yes—
133C2×Q8×C83C2×D411yescharacteristic
134C2×Q8×C83C2413yes—
135C2×Q8×C83C2416yes—
136C2×D4×C83—23no—
137C2×D4×C83—23no—
138C2×D4×C83—23no—
139C2×D4×C83C2413yes—
140C22×C4×C83—21no—
141C22×C4×C83—23no—
142C22×C4×C83—23no—
143C22×C4×C83—23no—
144C22×C4×C83—23no—
145C22×C4×C83C2×D413yes—
146C22×C4×C83C2×D413yes—
147C22×C4×C83C2411yescharacteristic
148Q16×C83—212no—
149Q16×C83—412no—
150SD16×C83—212no—
151SD16×C83—412no—
152M16×C83—26no—
153M16×C83#4114yescomplemented
154C2×C8×C83C2416yes—
155C2×C8×C83C2412yes—
156C4⋊C4×C83—26no—
157C4⋊C4×C83—26no—
158C4⋊C4×C83—26no—
159C4⋊C4×C83—26no—
160C4⋊C4×C83—26no—
161C4⋊C4×C83—26no—
162C4⋊C4×C83—21no—
163C4⋊C4×C83—23no—
164C4⋊C4×C83—23no—
165C4⋊C4×C83—23no—
166C4⋊C4×C83—23no—
167C4⋊C4×C83C2×D413yes—
168C4⋊C4×C83C2×D413yes—
169C4⋊C4×C83C2×D416yes—
170C4⋊C4×C83C2412yes—
171C4⋊C4×C83C2411yescharacteristic
172C4⋊C4×C83C2411yescharacteristic
173C22⋊C4×C83—21no—
174C22⋊C4×C83—26no—
175C22⋊C4×C83—23no—
176C22⋊C4×C83—23no—
177C22⋊C4×C83—26no—
178C42×C83—21no—
179C42×C83—23no—
180C42×C83—23no—
181C42×C83—26no—
182C42×C83—26no—
183C42×C83C2×D413yes—
184C42×C83C2×D413yes—
185C42×C83C2413yes—
186C42×C83C2413yes—
187C23×C83—23no—
188C23×C83C22×D411yescharacteristic
189Q8×C83—26no—
190Q8×C83—26no—
191Q8×C83—23no—
192Q8×C83—26no—
193Q8×C83—23no—
194Q8×C83—46no—
195Q8×C83#13714yescomplemented
196D4×C83—23no—
197D4×C83—23no—
198D4×C83—46no—
199D4×C83—46no—
200D4×C83—83no—
201C2×C4×C83—26no—
202C2×C4×C83—22no—
203C2×C4×C83—21no—
204C2×C4×C83—21no—
205C2×C4×C83—23no—
206C2×C4×C83—26no—
207C2×C4×C83—43no—
208C2×C4×C83—42no—
209C2×C4×C83—46no—
210C2×C4×C83—46no—
211C2×C4×C83—46no—
212C2×C4×C83—83no—
213C2×C4×C83C22×D413yes—
214C2×C4×C83C22×D413yes—
215C2×C4×C83C22×D413yes—
216C2×C4×C83C22×D411yescharacteristic
217C2×C4×C83C22×D413yes—
218C2×C4×C83C2×#4112yes—
219C2×C4×C83H(5,𝔽2)12yes—
220C2×C4×C83C2511yescharacteristic
221C8×C83—26no—
222C8×C83C2×#4114yes—
223C22×C83—21no—
224C22×C83—21no—
225C22×C83—43no—
226C22×C83—43no—
227C22×C83—43no—
228C22×C83#55012yes—
229C22×C83C23×D411yescharacteristic
230C4×C83—23no—
231C4×C83—23no—
232C4×C83—23no—
233C4×C83—42no—
234C4×C83—41no—
235C4×C83—43no—
236C4×C83—46no—
237C4×C83—41no—
238C4×C83#55012yes—
239C4×C83C2×#13716yes—
240C4×C83C22×#4111yescharacteristic
241C4×C83C2×H(5,𝔽2)11yescharacteristic
242#61404C8311yesmaximal, characteristic, direct product factor, Sylow
243C2×C83—22no—
244C2×C83—83no—
245C2×C83C2×#55011yescharacteristic
246C2×C83C22×#13711yescharacteristic
247C2×C83#358611yescharacteristic
248#3480C2×C8311yescharacteristic
249C2×#557C2×C8311yescharacteristic, complemented
250#3400C2×C8316yescomplemented
251#3398C2×C8316yescomplemented
252#3395C2×C8312yescomplemented
253#3392C2×C8312yescomplemented
254#3152C2×C8313yescomplemented
255#3039C2×C8313yescomplemented
256#3037C2×C8313yescomplemented
257Q8×M16C2×C8311yescharacteristic, complemented
258C4×#137C2×C8313yescomplemented
259C83#6140411yescharacteristic, direct product factor, Sylow
260C2×#137C22×C8313yes—
261C2×#137C22×C8316yes—
262#559C22×C8313yes—
263#557—21no—
264#557—23no—
265#557C22×C8312yes—
266#557C22×C8311yescharacteristic
267#557C22×C8316yes—
268#556C22×C8313yes—
269#553—23no—
270#550C22×C8311yescharacteristic
271#548—21no—
272#532C22×C8313yes—
273C2×C4⋊Q8C22×C8313yes—
274#529—23no—
275C2×#125—23no—
276C4×#41—21no—
277C4×#41C22×C8313yes—
278C2×C4×Q8—21no—
279C2×C4×Q8C22×C8313yes—
280C2×C4×Q8C22×C8311yescharacteristic
281#497C22×C83112yes—
282#494C22×C8316yes—
283#492C22×C8316yes—
284#464C22×C83112yes—
285#462C22×C8316yes—
286#462C22×C8316yes—
287#462C22×C8316yes—
288#461C22×C8316yes—
289#461C22×C8316yes—
290#460C22×C8316yes—
291#446C22×C8316yes—
292C8×Q8C22×C8312yes—
293#441C22×C8316yes—
294#440C22×C8316yes—
295C4×Q16C22×C8316yes—
296C4×SD16C22×C8316yes—
297#428C22×C8311yescharacteristic
298#423C22×C8313yes—
299#419C22×C8313yes—
300#417C22×C8313yes—
301C4×M16C22×C8313yes—
302C2×#41—21no—
303C2×#41C23×C8313yes—
304C22×Q8D4×C8311yescharacteristic
305C22×Q8C23×C8313yes—
306#137—212no—
307#137—412no—
308C2×Q16C23×C8316yes—
309C2×Q16C23×C83112yes—
310C2×SD16C23×C8316yes—
311C2×SD16C23×C83112yes—
312C2×M16C23×C8313yes—
313C2×M16C23×C8311yescharacteristic
314C4⋊Q8—23no—
315C4⋊Q8—23no—
316C4⋊Q8C23×C8316yes—
317C4⋊Q8C23×C8316yes—
318C4⋊Q8C23×C8313yes—
319#127C23×C8313yes—
320#126—26no—
321#125—26no—
322#125—26no—
323#125—212no—
324#125—23no—
325#124—23no—
326#124C23×C8316yes—
327#123—26no—
328C22⋊Q8—23no—
329#121—23no—
330#121—23no—
331#121—23no—
332C4×Q8—26no—
333C4×Q8—22no—
334C4×Q8—21no—
335C4×Q8—23no—
336C4×Q8—23no—
337C4×Q8D4×C8311yescharacteristic
338C4×Q8D4×C8313yes—
339C4×Q8C23×C8316yes—
340C4×Q8C23×C8316yes—
341C4×Q8C23×C8313yes—
342C4×Q8C23×C8313yes—
343C4×Q8C23×C8311yescharacteristic
344C4×Q8C23×C8313yes—
345C4×D4—23no—
346C4×D4—23no—
347C4×D4—23no—
348C4×D4—23no—
349C4×D4—23no—
350C4×D4C23×C8313yes—
351C4×D4C23×C8313yes—
352#117—23no—
353#117—23no—
354#117C23×C8311yescharacteristic
355C2×C4⋊C4—23no—
356C2×C4⋊C4—23no—
357C2×C4⋊C4—23no—
358C2×C4⋊C4D4×C8313yes—
359C2×C4⋊C4C23×C8311yescharacteristic
360C2×C42—23no—
361C2×C42—23no—
362C2×C42C23×C8313yes—
363#107C23×C8312yes—
364#106C23×C8312yes—
365C4⋊C8C23×C8316yes—
366C4⋊C8C23×C8316yes—
367#103C23×C8316yes—
368#103C23×C8316yes—
369#103C23×C8316yes—
370#102C23×C8316yes—
371#97C23×C8316yes—
372C4×C8C23×C8316yes—
373#41—26no—
374#41—26no—
375#41—44no—
376C2×Q8—24no—
377C2×Q8—23no—
378C2×Q8—26no—
379C2×Q8—26no—
380C2×Q8C2×D4×C8313yes—
381C2×Q8C2×D4×C8311yescharacteristic
382C2×Q8C24×C8313yes—
383C2×Q8C24×C8316yes—
384C2×D4—23no—
385C2×D4—23no—
386C2×D4—23no—
387C2×D4C24×C8313yes—
388C22×C4—21no—
389C22×C4—23no—
390C22×C4—23no—
391C22×C4—23no—
392C22×C4—23no—
393C22×C4C2×D4×C8313yes—
394C22×C4C2×D4×C8313yes—
395C22×C4C24×C8311yescharacteristic
396Q16—212no—
397Q16—412no—
398SD16—212no—
399SD16—412no—
400M16—26no—
401M16#41×C8314yescomplemented
402C2×C8C24×C8316yes—
403C2×C8C24×C8312yes—
404C4⋊C4—26no—
405C4⋊C4—26no—
406C4⋊C4—26no—
407C4⋊C4—26no—
408C4⋊C4—26no—
409C4⋊C4—26no—
410C4⋊C4—21no—
411C4⋊C4—23no—
412C4⋊C4—23no—
413C4⋊C4—23no—
414C4⋊C4—23no—
415C4⋊C4C2×D4×C8313yes—
416C4⋊C4C2×D4×C8313yes—
417C4⋊C4C2×D4×C8316yes—
418C4⋊C4C24×C8312yes—
419C4⋊C4C24×C8311yescharacteristic
420C4⋊C4C24×C8311yescharacteristic
421C22⋊C4—21no—
422C22⋊C4—26no—
423C22⋊C4—23no—
424C22⋊C4—23no—
425C22⋊C4—26no—
426C42—21no—
427C42—23no—
428C42—23no—
429C42—26no—
430C42—26no—
431C42C2×D4×C8313yes—
432C42C2×D4×C8313yes—
433C42C24×C8313yes—
434C42C24×C8313yes—
435C23—23no—
436C23C22×D4×C8311yescharacteristic
437Q8—26no—
438Q8—26no—
439Q8—23no—
440Q8—26no—
441Q8—23no—
442Q8—46no—
443Q8#137×C8314yescomplemented
444D4—23no—
445D4—23no—
446D4—46no—
447D4—46no—
448D4—83no—
449C2×C4—26no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—23no—
454C2×C4—26no—
455C2×C4—43no—
456C2×C4—42no—
457C2×C4—46no—
458C2×C4—46no—
459C2×C4—46no—
460C2×C4—83no—
461C2×C4C22×D4×C8313yes—
462C2×C4C22×D4×C8313yes—
463C2×C4C22×D4×C8313yes—
464C2×C4C22×D4×C8311yescharacteristic
465C2×C4C22×D4×C8313yes—
466C2×C4C2×#41×C8312yes—
467C2×C4H(5,𝔽2)×C8312yes—
468C2×C4C25×C8311yescharacteristic
469C8—26no—
470C8C2×#41×C8314yes—
471C22—21no—
472C22—21no—
473C22—43no—
474C22—43no—
475C22—43no—
476C22#550×C8312yes—
477C22C23×D4×C8311yescharacteristic
478C4—23no—
479C4—23no—
480C4—23no—
481C4—41no—
482C4—42no—
483C4—41no—
484C4—43no—
485C4—46no—
486C4#550×C8312yes—
487C4C2×#137×C8316yes—
488C4C22×#41×C8311yescharacteristic
489C4C2×H(5,𝔽2)×C8311yescharacteristic
490C2—22no—
491C2—83no—
492C2C2×#550×C8311yescharacteristic
493C2C22×#137×C8311yescharacteristic
494C2C83×#358611yescharacteristic
subgroupG/H
center, socleC22×C83C23×D4
commutator, Frattini, cosocleC2×C4C25×C83
prime radicalC83×#3480C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C83 ◁ C2×C4×Q8×C83 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#557×C83C2semidirect
3. C83×#3400C2semidirect
4. C83×#3398C2semidirect
5. C83×#3395C2semidirect
6. C83×#3392C2semidirect
7. C83×#3152C2semidirect
8. C83×#3039C2semidirect
9. C83×#3037C2semidirect
10. Q8×M16×C83C2semidirect
11. C4×#137×C83C2semidirect
153. M16×C83#41semidirect
195. Q8×C83#137semidirect
242. #61404C83direct (N × H)
249. C2×#557C2×C83semidirect
250. #3400C2×C83semidirect
251. #3398C2×C83semidirect
252. #3395C2×C83semidirect
253. #3392C2×C83semidirect
254. #3152C2×C83semidirect
255. #3039C2×C83semidirect
256. #3037C2×C83semidirect
257. Q8×M16C2×C83semidirect
258. C4×#137C2×C83semidirect
259. C83#61404direct (N × H)
401. M16#41×C83semidirect
443. Q8#137×C83semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C83×#34802
C2×#557×C832
C83×#34002
C83×#33982
C83×#33952
C83×#33922
C83×#31522
C83×#30392
C83×#30372
Q8×M16×C832
C4×#137×C832
#6140483